Chaos in a Nonautonomous Model for the Interactions of Prey and Predator With Effect of Water Level Fluctuation

Abstract

Water level regulates the dynamics of different populations residing in water bodies. The increase/decrease in the level of water leads to an increase/decrease in the volume of water, which influences the interactions of fishes and catching capability. We examine how seasonal variations in water level and harvesting affect the outcome of prey–predator interactions in an artificial lake. A seasonal variation of the water level is introduced in the predation rate. We derive conditions for the persistence and extinction of the populations. Using the continuation theorem, we determine the conditions for which the system has a positive periodic solution. The existence of a unique globally stable periodic solution is also presented. Moreover, we obtain conditions for the existence, uniqueness and stability of a positive almost periodic solution. We find that if the autonomous system has a stable focus, the corresponding nonautonomous system exhibits a unique stable positive periodic solution. But, whenever the autonomous system shows limit cycle oscillations, the corresponding nonautonomous system exhibits chaotic dynamics. The chaotic behavior of system is confirmed by the positivity of the maximal Lyapunov exponent. For higher values of the assimilation fraction of prey population, the persistent oscillations of the autonomous system are eliminated and this system becomes stable. On the other hand, chaotic nature of the nonautonomous system is converted into periodicity if the assimilation fraction of prey is large. Moreover, populations behave almost periodically if the seasonally varied rate parameters are almost periodic functions of time. Our findings show that water level plays an important role in the persistence of prey–predator system.

A nonautonomous model for the effect of rarity value on the dynamics of a predator–prey system with variable harvesting

“…The benefit of considering the above forms of q(t) and l(t) is that these time dependent parameters q(t) and l(t) have high and low seasons both in the periods where sin(𝜔t) is positive and negative, respectively [52,53]. Here, the parameters q 1 (0 < q 1 < q) and l 1 (0 < l 1 < l) decide the strength of seasonal forcing in q(t) and l(t), respectively.…”

“…We consider the sinusoidal forms of $$$ q(t) $$$ and $$$ l(t) $$$: $$$$ q(t)&amp;amp;amp;#x0003D;q&amp;amp;amp;#x0002B;{q}_1\sin \left(\omega t\right),l(t)&amp;amp;amp;#x0003D;l&amp;amp;amp;#x0002B;{l}_1\sin \left(\omega t\right). $$$$ The benefit of considering the above forms of $$$ q(t) $$$ and $$$ l(t) $$$ is that these time dependent parameters $$$ q(t) $$$ and $$$ l(t) $$$ have high and low seasons both in the periods where $$$ \sin \left(\omega t\right) $$$ is positive and negative, respectively [52, 53]. Here, the parameters $$$ {q}_1\kern0.1em \left(0&amp;amp;lt;{q}_1&amp;amp;lt;q\right) $$$ and $$$ {l}_1\kern0.1em \left(0&amp;amp;lt;{l}_1&amp;amp;lt;l\right) $$$ decide the strength of seasonal forcing in …”

A systematic study of autonomous and nonautonomous predator–prey models for the combined effects of fear, refuge, cooperation and harvesting

“…As many environmental factors affecting the survival of species in the ecological community are seasonally forced, considerations of parameters in ecological models as periodic functions of time rather than constants would mimic comparatively more realistic situations [ 49 , 50 ]. For instance, seasonal variation appears in the birth rate of many species that is caused due to humidity, temperature, rainfall, abundance of food, change in daylight period, etc., [ 51 ].…”